SOLUTION: Find the values of x, y, and z so that matrix A = 1 2 x 3 0 y 1 1 z is invertible.

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Question 1164525: Find the values of x, y, and z so that matrix A =
1 2 x
3 0 y
1 1 z
is invertible.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
A square matrix is Invertible if and only if its determinant is non-zero.

%28matrix%283%2C3%2C%0D%0A1%2C+2%2C+x%2C%0D%0A3%2C+0%2C+y%2C%0D%0A1%2C+1%2C+z%29%29

We find its determinant

abs%28matrix%283%2C3%2C%0D%0A1%2C+2%2C+x%2C%0D%0A3%2C+0%2C+y%2C%0D%0A1%2C+1%2C+z%29%29%22%22=%22%221%2A0%2Az%2B2%2Ay%2A1%2Bx%2A3%2A1-x%2A0%2A1-2%2A3%2Az-1%2Ay%2A1%22%22=%22%220%2B2y%2B3x-0-6z-y%22%22=%22%223x%2By-6z

We can choose any values for x, y, z such that 

3x%2By-6z%3C%3E0

I'll arbitrary choose x=3, y=-2, z=1 (to make the determinant 1, so its
inverse will have all integer elements:

%28matrix%283%2C3%2C%0D%0A1%2C+2%2C+3%2C%0D%0A3%2C+0%2C+-2%2C%0D%0A1%2C+1%2C+1%29%29

Then its inverse is

%28matrix%283%2C3%2C%0D%0A2%2C+1%2C+-4%2C%0D%0A-5%2C+-2%2C+1%2C%0D%0A3%2C+1%2C+-6%29%29

Edwin